Cardiovascular Engineering and Technology

, Volume 4, Issue 4, pp 520–534 | Cite as

A Simplified Approach for Simultaneous Measurements of Wavefront Velocity and Curvature in the Heart Using Activation Times

  • Nachaat Mazeh
  • David E. Haines
  • Matthew W. Kay
  • Bradley J. Roth


The velocity and curvature of a wave front are important factors governing the propagation of electrical activity through cardiac tissue, particularly during heart arrhythmias of clinical importance such as fibrillation. Presently, no simple computational model exists to determine these values simultaneously. The proposed model uses the arrival times at four or five sites to determine the wave front speed (v), direction (θ), and radius of curvature (ROC) (r 0). If the arrival times are measured, then v, θ, and r 0 can be found from differences in arrival times and the distance between these sites. During isotropic conduction, we found good correlation between measured values of the ROC r 0 and the distance from the unipolar stimulus (r = 0.9043 and p < 0.0001). The conduction velocity (m/s) was correlated (r = 0.998, p < 0.0001) using our method (mean = 0.2403, SD = 0.0533) and an empirical method (mean = 0.2352, SD = 0.0560). The model was applied to a condition of anisotropy and a complex case of reentry with a high voltage extra stimulus. Again, results show good correlation between our simplified approach and established methods for multiple wavefront morphologies. In conclusion, insignificant measurement errors were observed between this simplified approach and an approach that was more computationally demanding. Accuracy was maintained when the requirement that ε (ε = b/r 0, ratio of recording site spacing over wave fronts ROC) was between 0.001 and 0.5. The present simplified model can be applied to a variety of clinical conditions to predict behavior of planar, elliptical, and reentrant wave fronts. It may be used to study the genesis and propagation of rotors in human arrhythmias and could lead to rotor mapping using low density endocardial recording electrodes.


Cardiac muscle Propagation velocity Wave front curvature Electrode Anisotropy 



Wave front speed


Angle specifying wave front velocity direction


Radius of curvature


Recording sites spacing (shortest distance)


Ratio of electrode spacing over radius of curvature


Intracellular conductivity in the x-direction


Intracellular conductivity in the y-direction


Extracellular conductivity in the x-direction


Extracellular conductivity in the y-direction


Activation time at electrode n, where n = 1, 2, 3, or 4


Difference of activation times between the ith and jth electrodes


Time for wave front to travel segment i


Stimulation protocol using stimulus of strength S1 and at a later time stimulus S2


Side of square inside the tissue where fibers curve


Point of interest


Distance formula


Line segments method


Our computational method


Ventricular fibrillation


Membrane potential



This research was funded by the Department of Cardiovascular Medicine at Beaumont Health System, Royal Oak, Michigan. Dr. M. W. Kay received support from the NIH Grant (HL095828). We wish to thank Drs. R. A. Gray and J. M. Rogers for their helpful discussions and insights.

Conflict of interest


Supplementary material

13239_2013_158_MOESM1_ESM.doc (1.3 mb)
Supplementary material 1 (DOC 1283 kb)


  1. 1.
    Allessie, M. A., F. J. M. Bonke, and F. J. G. Schopman. Circus movement in rabbit atrial muscle as a mechanism of tachycardia, III: the “leading circle” concept. A new model of circus movement in cardiac tissue without involvement of an anatomical obstacle. Circ. Res. 41:9–18, 1977.CrossRefGoogle Scholar
  2. 2.
    Bayly, P. V., B. H. KenKnight, J. M. Rogers, R. E. Hillsley, R. E. Ideker, and W. M. Smith. Estimation of conduction velocity vector fields from epicardial mapping data. IEEE Trans. Biomed. Eng. 45:563–571, 1998.CrossRefGoogle Scholar
  3. 3.
    Beeler, G. W., and H. Reuter. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. 268:177–210, 1977.Google Scholar
  4. 4.
    Berbari, E. J., P. Lander, B. J. Scherlag, R. Lazara, and D. B. Gesesowitz. Ambiguities of epicardial mapping. J. Electrocardiol. 24:16–20, 1992.CrossRefGoogle Scholar
  5. 5.
    Berenfeld, O., and A. M. Pertsov. Dynamics of intramural scroll waves in three dimensional continuous myocardium with rotational anisotropy. J. Theor. Biol. 199:383–394, 1999.CrossRefGoogle Scholar
  6. 6.
    Cabo, C., A. M. Pertsov, W. T. Baxter, J. M. Davidenko, R. A. Gray, and J. Jalife. Wave-front curvature as a cause of slow conduction and block in isolated cardiac muscle. Circ. Res. 75:1014–1028, 1994.CrossRefGoogle Scholar
  7. 7.
    Charteris, N., and B. J. Roth. How hyperpolarization and recovery of excitability affect propagation through a virtual anode in the heart. Comput. Math. Methods Med. 2011:375059, 2011.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Davidenko, J. M., A. V. Pertsov, R. Salomonsz, W. Baxter, and J. Jalife. Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355:349–351, 1992.CrossRefGoogle Scholar
  9. 9.
    El-Sherif, N., E. B. Caref, H. Yin, and M. Restivo. The electrophysiological mechanism of ventricular arrhythmias in the long QT syndrome: tridimensional mapping of activation and recovery times. Circ. Res. 79:474–492, 1996.CrossRefGoogle Scholar
  10. 10.
    Ershler, P. R., and R. L. Lux. Derivative mapping in the study of activation sequence during ventricular arrhythmias. In: Proceedings of Computers in Cardiology, edited by K. L. Ripley. New York: IEEE Computer Society Press, 1986, pp. 623–624.Google Scholar
  11. 11.
    Fast, V. G., and A. G. Kleber. Role of wavefront curvature in propagation of cardiac impulse. Cardiovasc. Res. 33:258–271, 1997.CrossRefGoogle Scholar
  12. 12.
    Girouard, S. D., J. M. Pastore, K. R. Laurita, K. W. Gregory, and D. S. Rosenbaum. Optical mapping in a new guinea pig model of ventricular tachycardia reveals mechanisms for multiple wavelengths in a single reentrant circuit. Circulation 93:603–613, 1996.CrossRefGoogle Scholar
  13. 13.
    Gray, R. A., and J. Jalife. Mechanisms of cardiac fibrillation. Science 270:1222–1223, 1995.CrossRefGoogle Scholar
  14. 14.
    Gray, R. A., A. M. Pertsov, and J. Jalife. Spatial and temporal organization during cardiac fibrillation. Nature 392:75–78, 1998.CrossRefGoogle Scholar
  15. 15.
    Horner, S. M., Z. Vespalcova, and M. J. Lab. Electrode for recording direction of activation, conduction velocity, and monophasic action potential of myocardium. Am. J. Physiol. 272:H-1917–H-1927, 1997.Google Scholar
  16. 16.
    Ideker, R. E., W. M. Smith, S. M. Blanchard, S. L. Reiser, and E. V. Simpson. The assumptions of isochronal cardiac mapping. PACE 12:456–478, 1989.CrossRefGoogle Scholar
  17. 17.
    Janse, M. J., F. J. L. van Capelle, H. Morsink, A. G. Kléber, F. Wilms-Schopman, R. Cardinal, C. N. d’Alnoncourt, and D. Durrer. Flow of “injury” current and patterns of excitation during early ventricular arrhythmias in acute regional myocardial ischemia in isolated porcine and canine hearts: evidence for two different arrhythmic mechanisms. Circ. Res. 47:151–165, 1980.CrossRefGoogle Scholar
  18. 18.
    Kadish, A. H., J. F. Spear, J. H. Levine, R. F. Hanich, C. Prood, and E. N. Moore. Vector mapping of myocardial activation. Circulation 74:603–615, 1986.CrossRefGoogle Scholar
  19. 19.
    Kay, M. W., and R. A. Gray. Measuring curvature and velocity vector fields for waves of cardiac excitation in 2-D media. IEEE Trans. Biomed. Eng. 52:50–63, 2005.CrossRefGoogle Scholar
  20. 20.
    KenKnight, B. H., P. V. Bayly, R. J. Gerstle, D. L. Rollins, P. D. Wolf, W. M. Smith, and R. E. Ideker. Regional capture of fibrillating ventricular myocardium. Evidence of an excitable gap. Circ. Res. 77:849–855, 1995.CrossRefGoogle Scholar
  21. 21.
    Laxer, C., C. Alferness, W. M. Smith, and R. E. Ideker. The use of computer animation of mapped cardiac potentials in studying electrical conduction properties of arrhythmias. In: Proceedings of Computers in Cardiology, edited by A. Murray, and K. L. Ripley. Chicago, IL: IEEE Computer Society Press, 1990, pp. 23–26.Google Scholar
  22. 22.
    Lin, S.-F., R. A. Abbas, and J. P. Wikswo, Jr. High-resolution high-speed synchronous epifluorescence imaging of cardiac activation. Rev. Sci. Instrum. 68:213–217, 1997.CrossRefGoogle Scholar
  23. 23.
    Mazeh, N. The upper limit of vulnerability of the heart. PhD dissertation, Oakland University, Rochester, MI, 2008.Google Scholar
  24. 24.
    Mazeh, N., and B. J. Roth. A mechanism of the upper limit of vulnerability. Heart Rhythm 6:361–367, 2009.CrossRefGoogle Scholar
  25. 25.
    Moe, G. K., W. C. Rheinboldt, and J. A. Abildskov. A computer model of atrial fibrillation. Am. Heart J. 67:200–220, 1964.CrossRefGoogle Scholar
  26. 26.
    Pertsov, A. M., J. M. Davidenko, R. Salomonsz, W. T. Baxter, and J. Jalife. Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle. Circ. Res. 72:631–650, 1993.CrossRefGoogle Scholar
  27. 27.
    Pogwizd, S. M., and P. B. Corr. Reentrant and nonreentrant mechanisms contribute to arrhythmogenesis during early myocardial ischemia: results using three-dimensional mapping. Circ. Res. 61:352–371, 1987.CrossRefGoogle Scholar
  28. 28.
    Punske, B. B., Q. Ni, R. L. Lux, R. S. MacLeod, P. R. Ershler, T. J. Dustman, M. J. Allison, and B. Taccardi. Spatial methods of epicardial activation time determination in normal hearts. Ann. Biomed. Eng. 31:781–792, 2003.CrossRefGoogle Scholar
  29. 29.
    Rogers, J. M., M. Usui, B. H. KenKnight, R. E. Ideker, and W. M. Smith. A quantitative framework for analyzing epicardial activation patterns during ventricular fibrillation. Ann. Biomed. Eng. 25:749–760, 1997.CrossRefGoogle Scholar
  30. 30.
    Rosenbaum, D. S., and J. Jalife. Optical Mapping of Cardiac Excitation and Arrhythmias. Armonk, NY: Futura Pub Co, 2001.Google Scholar
  31. 31.
    Roth, B. J. How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle. J. Math. Biol. 30(6):633–646, 1992.CrossRefMATHGoogle Scholar
  32. 32.
    Roth, B. J. Electrical conductivity values used with the bidomain model of cardiac tissue. IEEE Trans. Biomed. Eng. 44:326–328, 1997.CrossRefGoogle Scholar
  33. 33.
    Smith, W. M., P. D. Wolf, E. V. Simpson, N. D. Danieley, and R. E. Ideker. Mapping ventricular fibrillation and defibrillation. In: Cardiac Mapping, edited by M. Shenesa, M. Borggrefe, and G. Breithard. Mount Kisco, NY: Futura Publishing Co, 1993, pp. 251–260.Google Scholar
  34. 34.
    Spach, M. S., W. T. Miller, III, D. B. Geselowitz, R. C. Barr, J. Kootsey, and E. A. Johnson. The discontinuous nature of propagation in normal canine cardiac muscle: evidence for recurrent discontinuities of intracellular resistance that affect the membrane currents. Circ. Res. 48:39–54, 1981.CrossRefGoogle Scholar
  35. 35.
    Spooner, P. M., R. W. Joyner, and J. Jalife. Discontinuous Conduction in the Heart. Armonk, NY: Futura Publishing, 1997.Google Scholar
  36. 36.
    Tung, L. A bi-domain model for describing ischemic myocardial D-C potentials. PhD Thesis, MIT, Cambridge, MA, 1977.Google Scholar
  37. 37.
    Winfree, A. T. Heart muscle as a reaction-diffusion medium: the roles of electric potential diffusion, activation front curvature, and anisotropy. Int. J. Bifurcation Chaos 7:487–526, 1997.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Biomedical Engineering Society 2013

Authors and Affiliations

  • Nachaat Mazeh
    • 1
  • David E. Haines
    • 2
  • Matthew W. Kay
    • 3
  • Bradley J. Roth
    • 4
  1. 1.Department of Cardiovascular MedicineBeaumont Health SystemRoyal OakUSA
  2. 2.Department of Cardiovascular MedicineOakland University William Beaumont School of MedicineRoyal OakUSA
  3. 3.Department of Electrical and Computer EngineeringGeorge Washington UniversityWashingtonUSA
  4. 4.Department of PhysicsOakland UniversityRochesterUSA

Personalised recommendations